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Indispensability Arguments and Mathematical Explanation in Science

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Indispensability Arguments and Mathematical Explanation in Science Indispensability Arguments and Mathematical Explanation in Science A thesis submitted by Josephine Marie Salverda in partial fulfilment of the requirements for the degree of MPhil Stud in Philosophical Studies at University College London. 1 I, Josephine Marie Salverda, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. 2 Abstract The aim of this thesis is to find a way to undermine the indispensability argument for mathematical platonism. In chapter 1 I provide a brief survey of the indispensability argument, arguing that the explanatory indispensability argument is stronger than earlier forms of the argument. This is because it has less controversial premises, appealing neither to confirmational holism nor to a strong naturalism but rather to inference to the best explanation, a principle of inference which both sides in the indispensability debate are taken to accept. Hence I take the explanatory indispensability argument as my target. In chapter 2, I provide a more detailed account of the way in which inference to the best explanation, or IBE, is involved in the explanatory indispensability argument. I present two readings of the argument, rejecting the first reading and arguing that a second reading, which involves an instance of IBE, is the most plausible. Chapter 3 considers whether there are genuine cases of mathematical explanation in science, focusing on an explanation from evolutionary biology provided by Alan Baker. I draw on the biological literature to argue that there is some reason to doubt that Baker's explanation meets the conditions for a successful application of IBE. In chapter 4 I examine a number of restrictions on IBE recently suggested in the indispensability debate. Firstly, I argue that the indexing account suggests a reasonable restriction on IBE, but that proponents of the indexing account have not yet shown that this restriction is successful in undermining the explanatory indispensability argument. Secondly, I examine a restriction on IBE proposed by Pincock, arguing that this restriction is also unsuccessful in blocking the support of mathematical claims through IBE. Thirdly, I propose a restriction on IBE motivated by scientific practice and which, I argue, successfully undermines the explanatory indispensability argument. 3 Contents Introduction 6 Chapter 1: Indispensability arguments for mathematical platonism Introduction 8 1.1: Setting up the debate 9 1.2: Dispensability 13 1.3: Naturalism and confirmational holism 14 1.4: An explanatory indispensability argument 17 1.5: Two new premises 19 1.5.1: Dispensability again 19 1.5.2: Inference to the best explanation 20 Conclusion 21 Chapter 2: How is inference to the best explanation involved in the explanatory indispensability argument? Introduction 22 2.1: Inference to the Best Explanation 23 2.2: The No Miracles argument for scientific realism 24 2.3: Connecting the explanatory indispensability argument to IBE and scientific realism 27 2.3.1: Version 1 of the explanatory indispensability argument 28 2.3.2: Version 2 of the explanatory indispensability argument 31 2.4: Laying out the options 33 Conclusion 34 Chapter 3: Periodical cicadas and best explanation in science Introduction 35 3.1: Periodical cicadas 36 3.2: Science over philosophy? 38 3.3: Alternative explanations 40 3.4: An appeal to the biological literature 43 Conclusion 46 4 Chapter 4: Restrictions on inference to the best explanation Introduction 48 4.1: The indexing account 49 4.1.1: A restriction on IBE 50 4.1.2: (Only) an indexing role? 53 4.2: Pincock's restriction on IBE 58 4.2.1: Background 58 4.2.2: Pincock's restriction on IBE 61 4.3: A new restriction on IBE 69 Conclusion 74 Conclusion 76 Bibiography 78 5 Introduction In this thesis, I search for a way to undermine the indispensability argument for mathematical platonism, a philosophical position which holds that there are abstract mathematical objects. The main opponent of the indispensability argument is the nominalist, who claims that there are no mathematical objects. Although I aim to undermine an important argument for mathematical platonism, my thesis does not serve as an argument for nominalism, since platonism and nominalism could both be incorrect. Since my aim is to undermine the indispensability argument, it will be advantageous first to locate a strong form of the indispensability argument, by which I mean a version of the indispensability argument with the fewest and least controversial premises, so that my arguments will have maximum effect. In chapter 1 I provide a brief survey of the indispensability argument, which allows me to locate a form of the argument known as the explanatory indispensability argument, which does not appeal to confirmational holism or to a strong reading of naturalism. Since these premises are controversial, an argument that does without these premises is more difficult to undermine. The explanatory indispensability argument involves an implicit appeal to inference to the best explanation, a principle of inference which both sides in the indispensability debate are taken to accept. This suggests that the explanatory indispensability argument will be difficult for the nominalist to undermine. I will thus have located a version of the indispensability argument which is particularly difficult to undermine, and which I will take as my target in the rest of the thesis. Since the explanatory indispensability argument is the subject of much recent discussion in the literature, my thesis also functions as a survey of recent attempts to undermine the explanatory indispensability argument. In chapter 2, I will clarify the way in which inference to the best explanation, or IBE, is involved in the explanatory indispensability argument. In most formulations of the explanatory indispensability argument the appeal to IBE is implicit. However, since inference to the best explanation is taken to be central in the recent indispensability debate, it will be useful to understand the role played by IBE in the explanatory indispensability argument in more detail. Therefore I will first provide a basic account of inference to the best explanation. Then, I will examine an application of IBE in the No Miracles argument for scientific realism. Platonists sometimes appeal to scientific realism in support of the explanatory indispensability argument, because, as I will argue, accepting scientific realism involves endorsing inference to the best explanation. However, the talk of scientific realism is often misleading and I 6 will argue that it is best dropped. Hence I will motivate a version of the explanatory indispensability argument which does not rely on scientific realism, but involves an instance of inference to the best explanation. According to this argument, the existence of a mathematical explanation in science motivates the claim that we ought rationally to believe in the existence of mathematical entities. In chapter 3, therefore, I examine an example of mathematical explanation in science drawn from evolutionary biology. This example, provided by Alan Baker, has been widely discussed in the recent literature. I will argue that any attempts to show that Baker has not provided a genuine case of mathematical explanation in science must be made on a scientific basis, and, for this reason, I will argue that various nominalistically acceptable alternatives fail. I will appeal to the biological literature to argue that there is some evidence for a lack of consensus on Baker's explanation amongst biologists, which suggests that Baker's example needs further scientific defence. Nevertheless, since further examples of mathematical explanation in science have been provided in science, and I wish to avoid a case-by-case study of mathematical application in science, I will seek a more general strategy of undermining the explanatory indispensability argument. In chapter 4, I examine three restrictions on IBE, proposed in order to block the instance of IBE that is involved in the explanatory indispensability argument. First, I will consider the indexing account, which, I will argue, involves a reasonable restriction on IBE. However, I will argue that proponents of the indexing account have not yet shown that this restriction blocks the support of mathematical claims via IBE. I then examine a second restriction proposed by Pincock, which I will argue is also unsuccessful. Finally, I propose a restriction on IBE which is motivated by scientific practice, and which, I argue, is successful in blocking the instance of IBE involved in the explanatory indispensability argument. Therefore, I can claim to have succeeded in my aim to undermine the explanatory indispensability argument. 7 Chapter 1: Indispensability arguments for mathematical platonism Introduction In this chapter, I will give a basic account of indispensability arguments for mathematical platonism. I will look at the premises of such arguments and examine which of these premises are required for the argument to go through. My aim is to find a strong version of the indispensability argument, meaning a version which is difficult to undermine. In general, since less controversial premises help to make an argument stronger than controversial premises, I aim to isolate a form of indispensability argument with the fewest and least controversial premises. The idea is for each premise to be as 'weak' as possible, in the sense that the premise
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